The Qualitative Research Of Nonlinear Kirchhoff Type Elliptic Boundary Value Problems

This thesis focuses on the nonlinear Kirchhoff-type elliptic boundary value problems.These problems have important practical significance and broad application prospect in physics and biology.We mainly discuss the Kirchhoff-type elliptic boundary value problems?P₁?^P₃?see Introduction?involving critical Sobolev exponent,and investigate the related properties of positive solutions of these three kinds of problems.The main work is reflected in the following two aspects:?1?In this thesis,we study a class of Kirchhoff-type elliptic critical boundary value problems?P₁?involving double singular terms.There are three aspects of difficulties in solving this problem.Firstly,nonlocal term appears in the equation,and then the equation is no longer a pointwise identity.Secondly,Since the corresponding functional to?P₁?contains the critical Sobolev exponent,the equation does not satisfy the Palais-Smale compactness conditions.Thirdly,due to the appearance of the negative exponential term in?P₁?,the corresponding energy functional to the equation is not differentiable.Furhermore,the functional contains the Hardy singularity which causes us to fail to apply the standard variational method to deal with such problem directly.We overcome the above difficulties by using the Nehari manifold,the concentration compactness principle due to Lions,the Hardy inequality and the Ekeland variational principle.Finally,we prove the existence and multiplicity of positive solutions for the equation?P₁?under the appropriate conditions.To the best of our knowledge,these results are new.?2?We investigate the Kirchhoff-type elliptic systems with critical Sobolev exponent firstly.The characteristic of this system is both nonlocal and strong coupling terms,thus the energy functional does not satisfy the Palais-smale compactness condition.Secondly,we also discuss a class of Kirchhoff-type elliptic system?P₃?involving homogeneous abstract term.The problem is more difficult to be solved because of the appearance of nonlocal and critical homogeneous abstract terms in the system.This makes it impossible for us to apply the classical variational methods to solve these two kinds of problems.We establish the existence results of the positive solution of these two kinds of systems by applying the Nehari manifold,fibering maps,the Euler identity,the properties of homogeneous nonlinearity and critical point theory.The above results extend and improve the recent ones of some scholars.